Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

A versatile and inexpensive apparatus for rapid parallel synthesis on solid support: description and synthesis illustration

A new inexpensive and practical apparatus for solid-phase chemistry and parallel synthesis is described. This new apparatus fills an important void in the availability of portable tools for the synthesis of libraries of compounds in multi-milligram amounts. Individual reaction tube capacities range in size from 4 mL to 500 mL of operating liquid volume. Reaction blocks of 36 tubes x 4 mL or 24 tubes x 150 mL allow flexibility of operation. Insert tubes with frit ends function as filter sticks for resin wash and for maintenance of inert atmosphere. An electronic controller device connects to the reaction tubes for programmable entry of pulses of inert gas for resin mixing or vacuum for resin wash. The utility of this apparatus is illustrated by the synthesis of libraries based on 4-methaneamine imidazoles.